Bisectors, Medians, and Altitudes Section 5-1 Agenda: 11/30/11 Do Now Problem involving Isosceles and Equilateral Triangles Review Do Now Vocabulary: Perpendicular bisector, angle bisector, distance from a point to a line Mini Lesson: Using properties of Perpendicular bisectors and Angle Bisectors to solve problems Independent /Group work Practice problems Share-out Discussion of answers Wrap-Up/ Summary Writing Exercise Lesson Quiz Homework Review Class Notes, Castle Learning Relationships in Triangles Bisectors, Medians, and Altitudes Section 5-1 Mrs. Padilla Geometry Fall 2011

Objectives (W.O.W.) To identify and use perpendicular bisectors & angle bisectors in triangles To identify and use medians & altitudes in triangles

Vocabulary Perpendicular Bisectors Angle Bisectors Locus Equidistant Medians Altitudes Points of Concurrency

AIM:. How do we use properties of Perpendicular AIM: How do we use properties of Perpendicular bisectors and Angle Bisectors ? Do Now(5 mins)

Perpendicular bisector in Triangles Perpendicular bisector: A line or line segment that passes through the midpoint of a side of a triangle and is perpendicular to that side. Theorem 5-1-: Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment. Theorem 5-2-: Any point equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment.

Perpendicular bisector theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. If C is on the perpendicular bisector of AB, then CA = CB. IF C M B A THEN C ~ M B A

Converse of the Perpendicular bisector theorem If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of a segment.

Perpendicular bisector, cont… For every triangle there are 3 perpendicular bisectors The 3 perpendicular bisectors intersect in a common point named the circumcenter. In the picture to the right point K is the circumcenter.

Angle bisectors of a Triangle Angle bisector of a triangle: A segment that bisects an angle of a triangle and has one endpoint at a vertex of the triangle and the other endpoint at another point on the triangle. Theorem 5-3: Any point on the bisector of an angle is equidistant from the sides of the angle. Theorem 5-4: Any point on or in the interior of an angle and equidistant from the sides of an angle, lies on the bisector of the angle.

Angle bisector theorem If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle. IF THEN If m< 1 = m< 2, then BC = BD. C B B 1 1 A 2 A 2 D ~

Angle bisector, cont… For every triangle there are 3 angle bisectors. The 3 angle bisectors intersect in a common point named the incenter In the picture to the right, point I is the incenter.

Median and Altitude Median: A segment that connects a vertex of a triangle to the midpoint of the side opposite to that vertex. Every triangle has three medians. Altitude: A segment that has an endpoint at a vertex of a triangle and the other on the line opposite to that vertex, so that the segment is perpendicular to this line. Do example 1, page 239 Altitudes of an obtuse triangle Altitudes of a right triangle

Median A line segment whose endpoints are a vertex of a triangle and the midpoint of the side opposite the vertex. In the picture to the right, the blue line segment is the median.

Median, cont… For every triangle there are 3 medians The 3 medians intersect in a common point named the centroid In the picture to the right, point O is the centroid.

Altitudes A line segment from a vertex to the line containing the opposite side and perpendicular to the line containing that side. In the picture above, ∆ABC is an obtuse triangle & ∠ACB is the obtuse angle. BH is an altitude.

Altitudes, cont… For every triangle there are 3 altitudes The 3 altitudes intersect in a common point called the orthocenter. In the picture to the right, point H is the orthocenter.

Points of Concurrency Concurrent Lines 3 or more lines that intersect at a common point Point of Concurrency The point of intersection when 3 or more lines intersect. Type of Line Segments Point of Concurrency Perpendicular Bisectors Circumcenter Angle Bisectors Incenter Median Centroid Altitude Orthocenter

Points of Concurrency, cont… Facts to remember: The circumcenter of a triangle is equidistant from the vertices of the triangle. Any point on the angle bisector is equidistant from the sides of the angle (Converse of #3) Any point equidistant from the sides of an angle lies on the angle bisector. (Converse of #2) The incenter of a triangle is equidistant from each side of the triangle. The distance from a vertex of a triangle to the centroid is 2/3 of the median’s entire length. The length from the centroid to the midpoint is 1/3 of the length of the median.

Points of Concurrency, cont…

Example: Using Perpendicular bisector Use the diagram to find AB. In the diagram, AC is the perpendicular bisector of DB. Therefore AB = AD 8x = 5x + 12 3x = 12 x = 4 Since you were asked for AB, not just x: AB = 8x = 8 • 4 = 32 C D B 5x + 12 8x A

AIM:. How do we use properties of Perpendicular AIM: How do we use properties of Perpendicular bisectors and Angle Bisectors ? Example

AIM:. How do we use properties of Perpendicular AIM: How do we use properties of Perpendicular bisectors and Angle Bisectors ? Examples Does the information given in the diagram allow you to conclude that C is on the perpendicular bisector of AB?

AIM:. How do we use properties of Perpendicular AIM: How do we use properties of Perpendicular bisectors and Angle Bisectors ? Examples Does the information given in the diagram allow you to conclude that P is on the angle bisector of angle A? A A A

How can you tell if a ray or line segment is an angle bisector? AIM: How do we use properties of Perpendicular bisectors and Angle Bisectors ? Summary How can you tell if a ray or line segment is an angle bisector? How can you tell if a ray or line segment is a perpendicular bisector?

Facts to Remember and Memorize! 1. Perpendicular Bisectors 2. Angle Bisectors 3. Medians 4. Altitudes 1. …form right angles AND 2  lines segments 2. …form 2  angles 3. …form 2  line segments 4. … form right angles

Lesson Quiz 1. Does D lie on the perpendicular bisector of AIM: How do we use properties of Perpendicular bisectors and Angle Bisectors ? Lesson Quiz 1. Does D lie on the perpendicular bisector of Draw the diagram and answer the question

Review Class Notes Sec 5.1N and Sec 4.8R HomeWork AIM: How do we use properties of Perpendicular bisectors and Angle Bisectors ? HomeWork Review Class Notes Sec 5.1N and Sec 4.8R

The End (Finally!) Oh yeah! Do homework tonight and STUDY these notes that you just took on Section 5-1!